Consider a dynamical system defined by a non-autonomous ordinary differential equation
$$ \dot z = X(z,t)$$
where $X$ is periodic with respect to time of period $T > 0$. Consider the flow evaluated at $T$, i.e., the map $\Phi^X_{T,0} : M \to M$, where $M$ is a manifold (but let us take an open subset of $\mathbb R^n$) and $\Phi^X_{T,0}$ is the map that assigns to each $z \in M$ the value at time $T$ of the solution that starts from $z$ at time $0$. This map can be interpreted as a Poincaré map in the extended quotient phase space $M \times S^1_T$, $S^1_T$ being the one dimensional circle of length $T$. Let us suppose that $\Phi^X_{T,0}$ has a fixed point $\bar z$. I would like to find an explicit example of $X$ such that $\bar z$ is an equilibrium of the system. I have found the sufficient condition $X(\bar z,t)=0$ for every $t$, but I am still missing the explicit example. Can someone help me?
Let $M = S^1$, so the extended phase space is a torus. Take $X(z,t)= -\sin (z) \, (2+\sin (t))$ and $\bar z = 0$.